The 93rd annual meeting of the International Association of Applied Mathematics and Mechanics took place from 30th of May til 2nd of June in Dresden.

Raphael Kuess, who is group member of the Neptun Subproject OPT and works at the Humboldt University in Berlin, presented first results in his talk "Parameter identification in piezoelectricity based on all-at-once and reduced regularization“ as part of the session on optimization of differential equations. In his contribution, he explored approaches to a more efficient solution of the piezoelectric material parameter identification problem.

Talks with research closely related to the NEPTUN Project where held by Benjamin Jurgelucks (Parameter identification of piezoelectrics improved by neural networks) and Leander Claes (Inverse procedure for the identification of piezoelectric material parameters supported by dense neural networks), which both evaluated the possibility of including machine-learning-based method into the inverse procedure to determine piezoelectric material parameters.

## Abstracts

### Raphael Kuess: Parameter identification in piezoelectricity based on all-at-once and reduced regularization

Piezoelectric devices have become indispensible in almost all households, industry and medicine with an applications range from mobile phones and diesel fuel injectors to ultrasound imaging and dental scalers. Due to the upcoming switch to lead-free piezoceramics and the associated non-reproducible characterization of the material properties, the consistent and reproducible characterization of the material parameter set is of significant relevance for practical applications. In this talk we will investigate the parameter identification problem for the piezoelectric partial differential equation using regularization based on all-at-once and reduced iterative methods. The choice of these formulations is of particular interest for nonlinear models, which is especially noticeable in computational aspects. Therefore, the procedure is based on modeling and solving these inverse problems in these different settings by fitting simulated experimental data. Furthermore, numerical examples will be provided.

### Benjamin Jurgelucks: Parameter identification of piezoelectrics improved by neural networks

Accurately determining material parameters of piezoelectric materials is a demanding undertaking as some material parameters express only low sensitivity and are thus very hard to reconstruct as part of an inverse problem using classical methods alone. In recent years many advances have been made on this topic such as increasing the sensitivity of low-sensitivity parameters via optimal design of experiments and providing accurate and cheap gradient information via Algorithmic Differentiation. However, as classical gradient-based optimization methods for inverse problems only converge locally providing an accurate initial guess of the parameters is particularly important. Because of the curse of dimensionality using machine learning techniques to accurately determine the material parameters requires a huge amount of data und thus solutions to the governing partial differential equation. However, by only requiring approximately correct parameter values less data is needed. These approximately correct parameter values can instead be provided to classical methods for inverse problems as the initial guess. This greatly enhances the variety of different piezoelectric materials of which material parameters can now be identified.

### Leander Claes: Inverse procedure for the identification of piezoelectric material parameters supported by dense neural networks

The accurate knowledge of quantitative material parameters is a prerequisite for simulationdriven design processes of piezoelectric sensors and actuators. Due to the large number of parameters required to describe the mechanical, electrical, and coupling behaviour of theses materials, the identification procedure is especially challenging. In this contribution we aim to identify a full set of piezoelectric material parameters using a single disc-shaped sample. This is achieved by implementing an inverse measurement procedure based on matching the measured impedance of the physical sample with the output of a finite-element simulation model, the forward model. Because gradient-based, local optimisation is used for the identification process, an initial estimate for the parameters is required. For this initial value estimation, the forward model is inverted using a dense neural network. Synthetic training data for the neural network is generated by evaluating the forward model an adequate number of times with randomised material parameters. The network architecture is chosen so all directly measurable quantitate, i.e. the samples of the impedance in frequency domain as well as the density are the input of the network, while the material parameters are the output of the network. After training is concluded, the measured quantities are supplied to the neural network, which yields the initial estimates for the material parameters. A forward simulation using these results shows good agreement with the physical behaviour of the sample, enabling an efficient gradient based optimisation in the subsequent, final step of material parameter identification.